To send this article to your account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account.
Find out more about sending content to .

To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle.
Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

By using this service, you agree that you will only keep articles for personal use, and will not openly distribute them via Dropbox, Google Drive or other file sharing services.
Please confirm that you accept the terms of use.

A complete classification is given of pentavalent symmetric graphs of order
$30p$
, where
$p\ge 5$
is a prime. It is proved that such a graph
${\Gamma }$
exists if and only if
$p=13$
and, up to isomorphism, there is only one such graph. Furthermore,
${\Gamma }$
is isomorphic to
$\mathcal{C}_{390}$
, a coset graph of PSL(2, 25) with
${\sf Aut}\, {\Gamma }=\mbox{PSL(2, 25)}$
, and
${\Gamma }$
is 2-regular. The classification involves a new 2-regular pentavalent graph construction with square-free order.

Let
$\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}G$
be a graph and
${{\tau }}$
be an assignment of nonnegative thresholds to the vertices of
$G$
. A subset of vertices,
$D$
, is an irreversible dynamic monopoly of
$(G, \tau )$
if the vertices of
$G$
can be partitioned into subsets
$D_0, D_1, \ldots, D_k$
such that
$D_0=D$
and, for all
$i$
with
$0 \leq i \leq k-1$
, each vertex
$v$
in
$D_{i+1}$
has at least
$\tau (v)$
neighbours in the union of
$D_0, D_1, \ldots, D_i$
. Dynamic monopolies model the spread of influence or propagation of opinion in social networks, where the graph
$G$
represents the underlying network. The smallest cardinality of any dynamic monopoly of
$(G,\tau )$
is denoted by
$\mathrm{dyn}_{\tau }(G)$
. In this paper we assume that the threshold of each vertex
$v$
of the network is a random variable
$X_v$
such that
$0\leq X_v \leq \deg _G(v)+1$
. We obtain sharp bounds on the expectation and the concentration of
$\mathrm{dyn}_{\tau }(G)$
around its mean value. We also obtain some lower bounds for the size of dynamic monopolies in terms of the order of graph and expectation of the thresholds.

We estimate double sums
$$\begin{eqnarray}S_{{\it\chi}}(a,{\mathcal{I}},{\mathcal{G}})=\mathop{\sum }\limits_{x\in {\mathcal{I}}}\mathop{\sum }\limits_{{\it\lambda}\in {\mathcal{G}}}{\it\chi}(x+a{\it\lambda}),\quad 1\leq a with a multiplicative character
${\it\chi}$
modulo
$p$
where
${\mathcal{I}}=\{1,\dots ,H\}$
and
${\mathcal{G}}$
is a subgroup of order
$T$
of the multiplicative group of the finite field of
$p$
elements. A nontrivial upper bound on
$S_{{\it\chi}}(a,{\mathcal{I}},{\mathcal{G}})$
can be derived from the Burgess bound if
$H\geq p^{1/4+{\it\varepsilon}}$
and from some standard elementary arguments if
$T\geq p^{1/2+{\it\varepsilon}}$
, where
${\it\varepsilon}>0$
is arbitrary. We obtain a nontrivial estimate in a wider range of parameters
$H$
and
$T$
. We also estimate double sums
$$\begin{eqnarray}T_{{\it\chi}}(a,{\mathcal{G}})=\mathop{\sum }\limits_{{\it\lambda},{\it\mu}\in {\mathcal{G}}}{\it\chi}(a+{\it\lambda}+{\it\mu}),\quad 1\leq a and give an application to primitive roots modulo
$p$
with three nonzero binary digits.

In 1976, Wiegold asked if every finitely generated perfect group has weight 1. We introduce a new property of groups, finitely annihilated, and show that this might be a possible approach to resolving Wiegold’s problem. For finitely generated groups, we show that in several classes (finite, solvable, free), being finitely annihilated is equivalent to having noncyclic abelianisation. However, we also construct an infinite family of (finitely presented) finitely annihilated groups with cyclic abelianisation. We apply our work to show that the weight of a nonperfect finite group, or a nonperfect finitely generated solvable group, is the same as the weight of its abelianisation. This recovers the known partial results on the Wiegold problem: a finite (or finitely generated solvable) perfect group has weight 1.

Let
$\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}G$
be a finite group of order
$n$
, and let
$\text {C}_n$
be the cyclic group of order
$n$
. For
$g\in G$
, let
${\mathrm{o}}(g)$
denote the order of
$g$
. Let
$\phi $
denote the Euler totient function. We show that
$\sum _{g \in \text {C}_n} \phi ({\mathrm{o}}(g))\geq \sum _{g \in G} \phi ({\mathrm{o}}(g))$
, with equality if and only if
$G$
is isomorphic to
$\text {C}_n$
. As an application, we show that among all finite groups of a given order, the cyclic group of that order has the maximum number of bidirectional edges in its directed power graph.

It is a well-known result that if a nonconstant meromorphic function
$\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}f$
on
$\mathbb{C}$
and its
$l$
th derivative
$f^{(l)}$
have no zeros for some
$l\geq 2$
, then
$f$
is of the form
$f(z)=\exp (Az+B)$
or
$f(z)=(Az+B)^{-n}$
for some constants
$A$
,
$B$
. We extend this result to meromorphic functions of several variables, by first extending the classic Tumura–Clunie theorem for meromorphic functions of one complex variable to that of meromorphic functions of several complex variables using Nevanlinna theory.

In this paper, we consider the dependence of eigenvalues of sixth-order boundary value problems on the boundary. We show that the eigenvalues depend not only continuously but also smoothly on boundary points, and that the derivative of the
$\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}n$
th eigenvalue as a function of an endpoint satisfies a first-order differential equation. In addition, we prove that as the length of the interval shrinks to zero all higher eigenvalues of such boundary value problems march off to plus infinity. This is also true for the first (that is, lowest) eigenvalue.

We show that the null limit hypothesis, in the definition of a barrier, can be relaxed for normal boundary points that satisfy a mild additional condition. We also give a simple necessary and sufficient condition for the regularity of semi-singular boundary points.

Let
$\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}K$
be a locally compact hypergroup endowed with a left Haar measure and let
$L^1(K)$
be the usual Lebesgue space of
$K$
with respect to the left Haar measure. We investigate some properties of
$L^1(K)$
under a locally convex topology
$\beta ^1$
. Among other things, the semireflexivity of
$(L^1(K), \beta ^1)$
and of sequentially
$\beta ^1$
-continuous functionals is studied. We also show that
$(L^1(K), \beta ^1)$
with the convolution multiplication is always a complete semitopological algebra, whereas it is a topological algebra if and only if
$K$
is compact.

In this paper, we completely determine the commutativity of two Toeplitz operators on the harmonic Bergman space with integrable quasihomogeneous symbols, one of which is of the form
$\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}e^{ik\theta }r^{\, {m}}$
. As an application, the problem of when their product is again a Toeplitz operator is solved. In particular, Toeplitz operators with bounded symbols on the harmonic Bergman space commute with
$T_{e^{ik\theta }r^{\, {m}}}$
only in trivial cases, which appears quite different from results on analytic Bergman space in Čučković and Rao [‘Mellin transform, monomial symbols, and commuting Toeplitz operators’, J. Funct. Anal.154 (1998), 195–214].